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arXiv · 2502.01328

\'El\'ements de comptage sur les g\'en\'erateurs du groupe modulaire et les $\lambda$-quiddit\'es

Flavien Mabilat
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Abstract

The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_{1} & -1 \\[4pt] 1 & 0 \end{pmatrix}=\pm M$ when $M$ is equal to the generators of the modular group $S=\begin{pmatrix} 0 & -1 \\[4pt] 1 & 0 \end{pmatrix}$ and $T=\begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}$. To count these elements, we will study the $\lambda$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed.

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Flavien Mabilat. 2025-02-03. \'El\'ements de comptage sur les g\'en\'erateurs du groupe modulaire et les $\lambda$-quiddit\'es. https://arxiv.org/abs/2502.01328

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